## Derivative of amplitude and phase from a complex number It is a common trick to separate the real and imaginary part of a complex function or complex number. For a complex function $A(t) = R(t) e^{i\phi(t)}$, where $R(t)$ is the amplitude and $\phi(t)$ is the phase, one can write the derivative as $\mathrm{d}A = \frac{\mathrm{d}A}{\mathrm{d}t} \mathrm{d}t$ $\frac{\mathrm{d}A}{\mathrm{d}t} = \frac{d(R e^{i\phi})}{\mathrm{d}t} = \frac{dR}{\mathrm{d}t} e^{i\phi} + iR e^{i\phi} \frac{d\phi}{\mathrm{d}t}$ Consider $\frac{1}{A} \frac{\mathrm{d}A}{\mathrm{d}t}$, $\frac{1}{A} \frac{\mathrm{d}A}{\mathrm{d}t} = \frac{1}{R e^{i\phi}} (\frac{dR}{\mathrm{d}t} e^{i\phi} + iR e^{i\phi} \frac{d\phi}{\mathrm{d}t}) = \frac{1}{R} \frac{dR}{\mathrm{d}t} + i \frac{d\phi}{\mathrm{d}t}$ If extract the imaginary part only, we will have the derivative of phase $\frac{d\phi}{\mathrm{d}t} = \text{Im}\left(\frac{1}{R} \frac{dR}{\mathrm{d}t} + i \frac{d\phi}{\mathrm{d}t}\right) = \text{Im}\left(\frac{\frac{\mathrm{d}A}{\mathrm{d}t}}{A}\right)= \text{Im}\left(\frac{1}{A} \frac{\mathrm{d}A}{\mathrm{d}t}\right)$ Similarly, the derivative of amplitude is $\frac{dR}{\mathrm{d}t} = \text{Re}\left(\frac{\frac{\mathrm{d}A}{\mathrm{d}t}}{A}\right) R $